Answer to Question #157111 in Mechanics | Relativity for Terry

Question

The speed of light C is related to the permeability and thr permittivity e by the expression
C^2*uo*e = 1
(i) Show that this equation is homogenous
(ii) Calculate the value of uo

Answer 1

1. $[c^2] = \frac{m^2}{s^2}$, $[\mu_0] = \frac{H}{m}, [\epsilon_0] = \frac{F}{m}$, thus $[\mu_0 \epsilon_0] = \frac{F \cdot H}{m^2} = \frac{\frac{F}{C^2} \cdot HA^2 s^2}{m^2}$, as $C=A\cdot s$, but $C^2/F = HA^2 = J$ and thus $[\mu_0 \epsilon_0]= \frac{s^2}{m^2}, [c^2\mu_0 \epsilon_0]=1$ and so this equation is homogeneous.
2. From this identity we find $\mu_0 = \frac{1}{\epsilon_0 c^2} \approx \frac{1}{8.85\cdot 10^{-12}\cdot 9\cdot 10^{16}} = 1.26 \cdot 10^{-6} H/m$. Commonly the value of $\mu_0$ is approached by $4\pi \cdot 10^{-7} H/m$.